Bake the biscuits. Prepare and bake the biscuits according to your recipe or package directions. Keep them warm and set aside.
Cook the sausage (or bacon). In a large skillet over medium heat, cook the sausage patties until browned and cooked through, about 4-5 minutes per side (follow your package directions). Transfer to a paper towel-lined plate. If using bacon, cook until crispy in a skillet or on a sheet pan in the oven at 400°F for about 15-18 minutes.
Scramble the eggs. Whisk together the eggs, milk, salt, and pepper. Melt butter in a nonstick skillet over medium-low heat. Add egg mixture and cook gently, stirring often, until soft and just set, about 3 minutes. Remove from the heat while the eggs are still slightly glossy (they'll continue to set from residual heat).
Assemble the sandwiches. Split each biscuit in half. Spread an optional condiment on the biscuit halves if desired (such as a drizzle of honey, a dash of hot sauce, or a thin layer of mayo). Layer the bottom half with a sausage patty (or 2 slices of bacon), a portion of scrambled eggs, and a slice of cheese (tear or fold the cheese to fit if needed). Place the top biscuit half on top.
Serve warm. Serve immediately, or let the heat from the eggs melt the cheese for a minute or two before eating. If the cheese doesn't melt enough, wrap the sandwich loosely in foil for a minute or pop it in the microwave for 10-15 seconds.